A4 - 1D simple harmonic oscillator perturbed by a quartic potential

consider a 1D SHO perturbed by:

δ x^{3} + γ x^{4}

the first-order energy correction can be calculated as:

E_{0}^{1} = {(\frac{m ω}{π ℏ})}^{\frac{1}{2}} \int_{- \infty}^{\infty} γ x^{4} \exp (- \frac{m ω x^{2}}{ℏ}) d x = \frac{3 γ}{4} {(\frac{ℏ}{m ω})}^{2}

using standard integrals and the fact that the integral in $x^{3}$ is zero
the second-order shift of $E_{0}^{0}$ due to $ϕ_{1}^{0}$ with $E_{1}^{0} :$

E_{0}^{2} = - \frac{9 δ^{2} ℏ^{2}}{8 m^{3} ω^{4}}

using the standard integrals and the fact that the integral in $x^{5}$ is zero